House Speaker John A. Boehner praised tea partyers Thursday morning, and then members of the grass-roots movement assailed him in the afternoon, saying the Ohio Republican shouldn’t give in to Democrats‘ demands in the spending battle in Congress, even if that results in a government shutdown.

With anxiety growing as their electoral successes have not translated into major legislative gains, tea party leaders and their allies in Congress are warning lawmakers of all stripes that political penalties will result if they avoid making the tough decisions needed to put the country on a sustainable fiscal path.

Overall, the spending debate has strained the GOP’s relationship with the tea party, and nowhere is that more evident than with Mr. Boehner. The speaker says he values the insurgent movement’s role, but SenateDemocrats say that the tea party is the only thing preventing the speaker from making a deal that would cut spending and get near-universal approval from Congress.

The story line continued to play out Thursday when hundreds of tea partyers gathered across the street from the Capitol for a “Continuing Revolution” rally, sponsored by the Tea Party Patriots, where they waved signs that read “Grow a spine” and “Remember your promises, we do,” while calling on the Republicans to follow through on their campaign pledge to cut $100 billion in spending before Sept. 30, the end of the fiscal year.

“Cut it or shut it,” many in the crowd chanted, airing a lingering frustration that has been compounded by news reports suggesting that Republicans are hashing out the details of a deal with Democrats that would slice $33 billion in this year’s spending, a cut that falls far short of the party’s pledge.

“That’s nothing. It’s a pebble in a pool,” said Tom Jaremko, who traveled to the rally from Raleigh, N.C., with his wife, Diane. “They think we’re stupid.”

John Balazek, a metal worker from Maryland, said Mr. Boehner is “as lame and as weak a leader as you could ask for.”

“He’s caved and capitulated at every turn,” he said.

Others shared the frustration, but said Mr. Boehner has to weigh how the political fallout from a government shutdown would affect his party in the next election, where he hopes to increase his GOP numbers and make it easier to reduce federal spending over the long run.

“He has conflicting goals,” said James Renwick Manship, a tea party regular who was dressed in regalia reminiscent of the George Washington era. “One is certainly fixing the budget, and the other is winning the Senate [in 2012], so we can cut the budget and win the spending war” over the long haul.

Tea party leaders told The Washington Times this week that the spending battle will set the tone and that Mr. Boehner could face a primary challenge in the next election because he has not done enough to cut spending.

Despite the mounting criticism, Mr. Boehner, who began courting the tea party a year ago to help lay the groundwork for the Republicans’ November election wins, came to the defense of the movement at a news conference Thursday. He told reporters that the movement has an important role in the public debate.

“Listen, I’m glad that they’re here. And I’m glad that they’re engaged in the process. You know, I said over a year ago that we should talk to the tea party folks, that we should listen to them, and we should walk amongst them,” he said. “I don’t feel any differently about it today.”

Still, he made clear the limits of what he and his caucus can do.

“We control one-half of one-third of the government here in Washington,” he said. “We can’t impose our will on the Senate. All we can do is to fight for all of the spending cuts that we can get an agreement to.”

Asked how willing he would be to negotiate a deal that leaves behind conservatives, Mr. Boehner made clear where his allegiance lies: “Not very interested.”

Democratic leaders have banked on a divide-and-conquer strategy in spending negotiations with Republicans, as they try to drive a wedge between Mr. Boehner and tea party-aligned lawmakers, many of them freshmen who have little appetite for compromising on spending cuts.

“The tea party may have helped the Republicans win the last election, but they’re not helping the Republicans govern. The tea party is a negative force in these talks,” said Sen. Charles E. Schumer, New York Democrat, who has spoken regularly on the Senate floor this week in attempts to pressure Mr. Boehner to turn his back on tea partyers.

Mr. Schumer said the tea party’s engagement in politics is “as American as it gets,” but that their spending-cuts goals “are extreme because they’re out of step with what most Americans want” in terms of spending cuts.

Mr. Schumer unintentionally revealed at the start of a conference call this week that painting the tea party as “extreme” was a party talking point. “I always use the word ‘extreme’ — that is what the caucus instructed me to do the other week,” he said when he thought the call hadn’t started recording.

Responding to a spate of attacks against tea partyers, Senate Minority Leader Mitch McConnell, Kentucky Republican, told his colleagues that “the goals of the tea party sound pretty reasonable.”

“These folks recognize the gravity of the problems we face as a nation, and they’re doing something about it for the sake of our future,” he said. “They’re engaged in the debate about spending and debt — which is a lot more than we can say about the president and many Democrats in Congress.”